Thermoelectricity is the conversion from temperature differentials to electricity or vice versa. Thermoelectricity can be accomplished based on the Peltier-Seebeck effect, thermionic emission, or indirectly through magnetohydrodynamics.
Thermionic emission refers to the flow of electrons from a metal or metal oxide surface, caused by thermal vibrational energy overcoming the electrostatic forces holding electrons to atoms or molecules at the surface. This effect increases dramatically with increasing temperature (1000-3000 K), but is always present at temperatures above absolute zero. The science dealing with this phenomenon is thermionics. The charged particles are referred to as thermions.
Owen Willans Richardson worked with thermionic emission and received a Nobel prize in 1928 for his work on the thermionic phenomenon. Regarding Richardson's Law, in any metal, there are generally one or two electrons per atom that are free to move from atom to atom. This is sometimes referred to as a “sea of electrons”. Their velocities follow a statistical distribution, rather than being uniform, and occasionally an electron will have enough velocity to exit the metal without being pulled back in. The minimum amount of energy needed for an electron to leave the surface is referred to as the work function, and varies from metal to metal. A thin oxide coating is often applied to metal surfaces in vacuum tubes to provide a lower work function, as it is generally easier for electrons to leave the surface of the oxide. Richardson's Law, also called the Richardson-Dushmann equation, states that the emitted thermionic current density J (A/m2) is related to temperature T by the equation:J=AT2e−w/kT 
where T is the metal temperature in kelvins, W is the work function of the metal, and k is the Boltzmann constant. The proportionality constant A, known as Richardson's constant, given by
  A  =                    4        ⁢        π        ⁢                                  ⁢                  mk          2                ⁢        e                    h        3              =          1.20173      ×              10        6            ⁢              A                              m            2                    ⁢                      k            2                              
where m and e are the mass and charge of an electron, and h is Planck's constant. Because of the exponential function, the current increases rapidly with temperature when T is less than W. At higher temperatures the increase is quadratic in T.
The Richardson-Dushman equation must generally be corrected for the Schottky Effect. The current emitted from the metal cathode into the vacuum depends on the metal's thermionic work function, and that this function is lowered from its normal value by the presence of image forces and by the electric field at this cathode. This enhancement is given by the Field-enhanced thermionic emission (FEE) equation:J(Es,T,W)=AT2e−(W−ΔW)/kT ΔW=[eEc/(4π∈0)]1/2 
where Ec is the electric field strength at the cathode spot, and ∈0 is the vacuum permittivity. This equation is relatively accurate for electric field strengths lower than about 108 V m−1. For electric field strengths higher than 108V m−1 the use of the Murphy and Good equation for thermo-field (T-F) emission is more appropriate.
The conversion of heat to electricity by thermoelectric devices based on thermionics may play a key role in the future for energy production and utilization and thus begin replacing batteries. However, in order to meet that role, more efficient thermoelectric materials are needed that are suitable for high-temperature applications. Thermoelectric devices are generally based on heavily doped semiconductors and utilize transport through a vacuum, such as a hot electron emitter separated by a vacuum gap, or a vacuum gap filled with alkali metal gas such that alkali cations that act as counterions to improve the electron transport through the insulating gap region. Thermoelectric devices can be used for cooling applications or for electricity generation directly from a heat source.
A broad search has been under way to identify new materials with enhanced thermoelectric properties. Although the emphasis has been on finding materials that are superior to the well-known Bi2xSbxTe3ySey alloys used in cooling, interest in developing materials with high ZT values (high efficiency at high temperature) for direct energy conversion has been increasing. The figure of merit for the material of a thermionic converter is given as follows as a dimensionless quantity ZT as follows:ZT=(σS2/κ)T, 
where σ is the electrical conductivity of the converter from end to end, S is the thermopower or Seebeck coefficient of the converter, and K is the thermal conductivity of the material between the hot emitter and the cold collector side, and T is the temperature. The numerator (σS2) is referred to as the “power factor”. Several classes of materials are currently under investigation for use in thermionic converters, including complex chalcogenides, skutterudites, half-Heusler alloys, metal oxides, intermetallic clathrates, and pentatellurides. In addition, artificial superlattice thin-film structures grown from chemical vapor deposition, such as Bi2Te3/Sb2Te3, and by molecular beam epitaxy (MBE), such as PbSe0.98Te0.02/PbTe, have been introduced with substantially enhanced ZT values relative to those of their bulk counterparts. Marking an important development in this area, specially constructed Bi2Te3/Sb2Te3 superlattices were reported to exhibit a very high ZT of ˜2.4 at room temperature. The MBE-grown thin-film PbSe0.98Te0.02/PbTe systems feature peculiar pyramidal-shaped “nanodots” of PbSe that form spontaneously (surrounded by the higher band gap matrix material PbTe). The resulting samples possess a ZT of ˜2 at elevated temperatures (about 500 to 700 K). Nevertheless, because the vast majority of applications require materials in large (bulk) quantities, it would therefore be desirable to have compositions that generate similar ZT values in a bulk material.